The research will be directed to the field of theoretical and applied mechanics. Themes are:1* Nonlinear deterministic and stochastic processes and stability in dynamics of the complex systems and sandwich structures, as well as in the coupled rigid and solid bodies constrained by different constraints and its constitutive relations. 2* Mechanics and modelling of the behaviour and properties of the materials, which can be used in active mechanical structures. Behaviour of piezoelectric materials and their properties, which enable them to be integrated in the form of active elements into adaptive structures. The possibilities for the development of control laws, which will enable desired behaviour of the active materials and structures. The optimal control.3* Stability on finite and infinite time interval of the large discrete and continuous systems and evaluation of the practical stability.4* Integrity of dynamical systems and fracture mechanics. Evaluation of the state of mechanical systems. Global and local stress and strain state as well as strain potential energy in a material with crack under conditions of static or slowly changing dynamic loads. Stress analysis of composite structure.5* Models, dynamics and stability of vibroimpact systems. Nonlinear deterministic and stochastic processes in vibroimpact systems and stress and strain state and stability in structure and elements. Control oscillations in vibroimpact structure. Integrity of vibroimpact structures.
Today, it seems generally accepted that nonlinear vibrations problems should be cooperatively addressed through the combined use of analytical, computational, geometrical and experimental approach. Further, interaction between nonlinear dynamics and control plays an important role in advanced engineering systems in order to obtain desired dynamics behavior and improved reliability during operation.The nonlinear deterministic and stochastic dynamics and control of processes in mechanical systems are subject of our project research. New mathematical and phenomenological knowledge will be a advances to theoretical and applied mechanics of the rigid and solid bodies and complex sandwich structures with applications in mechanical engineering. Mechanics of materials with coupled fields are also subject of project research. Special attention will be paid to piezoelectric materials and their properties, which enable them to be integrated in the form of active elements into adaptive structures. The possibilities for the development of control laws, which will enable desired behaviour of the active materials and structures will be considered as well.Advances in the form of mathematical description of deterministic and stochastic dynamics of complex systems with coupled rigid and solid bodies by standard light element are planed as research results. Dynamics of the sandwich multibelt, multibeam or multiplate systems with rheological and or hereditary properties and deterministic and stochastic stability of the deformable shape are also subject of research.Using nonlinear and /or integro-differential relations or relations with fractional order derivatives the constitutive relations are described. Research results will be advances to the solving of the system of ordinary or partial integro-differential equations or fractional order differential equations with boundary and initial conditions. Proposed research are topics in the world scientific community, and are important for both advances to the mathematical theory of mechanics and for applications in engineering. Research will be directed to the field of theoretical and applied mechanics and realized into following research themes:1* Nonlinear deterministic and stochastic processes and stability in dynamics of the complex systems and sandwich structures, as well as in the coupled rigid and solid bodies constrained by different constraints and its constitutive relations. 2* Mechanics and modelling of the behaviour and properties of the materials, which can be used in active mechanical structures. Behaviour of piezoelectric materials and their properties. The development of control laws and optimal control.3* Stability on finite and infinite time interval of the large discrete and continuous systems and evaluation of the practical stability.4* Integrity of dynamical systems and fracture mechanics. Evaluation of the state of mechanical systems.5* Models, dynamics and stability of vibroimpact systems. Control oscillations in vibroimpact structure. Integrity of vibroimpact structures.Proposed research are the topics. It is visible from the contents of the main scientific journals and newly monographs in the field of the mechanics and nonlinear sciences and from the programs of the important world congresses as: ICTAM 2005, IUTAM Roma 2003, IFNA WCNA 2004, 11th IFToMM WC China 2004, EFC16-IDS 2006, ENOC 1999, 2002, 2005, IFAC FDA Bordeaux 2004, ICIAM 1999, GAMM 2005, ISUMEL Lvov 2005, YUCTAM 2005.
New mathematical knowledge of nonlinear deterministic and stochastic dynamics and control of processes in mechanical systems as a contribution to theoretical and applied mechanics of the rigid and solid bodies and complex sandwich structures with applications in mechanical engineering. New contribution to the mechanics of materials with rheological properties, with crack or with coupled fields. Contributions in the form of mathematical description of deterministic and stochastic dynamics of the sandwich multibelt, multibeam or multiplate systems with rheological or hereditary properties and stability of its deformable shape. Evaluation of the results: Publications of the scientific results in the scientific journals and publications of the scientific meeting in organization IUTAM, EuroMech and other. two or three monograph: Dynamic, elements and control of vibrations of the active structures and Theory of random vibrations Ph.D these and a number of Master theses.